Problem 2 - College Physics

Question

\(\bullet\) A rocket is moving to the right at half the speed of light relative to the earth. A lightbulb in the center of a room inside the rocket suddenly turns on. Call the light hitting the front end of the room event \(A\) and the light hitting the back of the room event \(B\) . (See Figure \(27.23 . )\) Which event occurs first, \(A\) or \(B\) , or are they simultaneous, as viewed by (a) an astronaut riding in the rocket and (b) a person at rest on the earth?

Step-by-Step Solution

Answer

(a) Simultaneous for the astronaut; (b) Event B occurs before event A for the Earth observer.

Step 1: Understand the Scenario

The light bulb is positioned at the center of the rocket room. When the light bulb turns on, light travels to both ends of the room.

Step 2: Analyze from Astronaut's Perspective

In the reference frame of the astronaut (who is at rest relative to the rocket), the light will reach the front and back ends simultaneously since the speeds of light are equal in both directions within this inertial frame.

Step 3: Analyze from Earth Observer's Perspective

From the perspective of the observer on Earth, the rocket is moving to the right at half the speed of light. Due to this motion and the invariance of the speed of light, the light moving towards the back of the rocket (against the direction of motion) will reach the back end first, before the light moving towards the front (in the same direction as the motion).

Step 4: Conclude on Event Occurrence

For the astronaut in the rocket, events A and B are simultaneous. For the observer on Earth, event B occurs before event A.

Key Concepts

Speed of Light

The speed of light in vacuum, often denoted as "c", is a fundamental constant in physics. It is approximately equal to 299,792,458 meters per second. A remarkable thing about the speed of light is its invariance—it remains constant for all observers, regardless of their relative motion.
This concept is counterintuitive because in everyday experiences, the speed of objects seems to vary depending on how fast we're moving relative to them.

• The speed of light is the same in all inertial frames of reference.
• No object with mass can reach or exceed this speed.
• Light's constant speed leads to fascinating concepts in physics such as time dilation and length contraction.

The rocket example provides a great context to appreciate this fundamental property. Whether viewed from someone on the rocket or someone on Earth, the light's speed doesn't change.

Inertial Reference Frame

An inertial reference frame is one in which an object not subject to forces moves at a constant speed in a straight line. In simple terms, it's a frame of reference that is either at rest or moving at a constant velocity.
These frames are crucial in understanding relativity because they provide simplified conditions where the laws of physics remain consistent.

• Newton's laws of motion are applicable in inertial frames without needing additional corrections.
• Relativity assumes these frames to find out how different observers view the same events differently.

In the rocket scenario, the rocket itself (along with the astronaut inside) is described as an inertial reference frame because, to the astronaut, everything inside moves uniformly unless acted upon by an external force like turning on the light.

Simultaneity in Relativity

Simultaneity in relativity refers to the concept that two events considered to occur at the same time in one reference frame may not be simultaneous in another. This is one of the core insights of Einstein's theory of relativity.
In the case of the rocket:

• For the astronaut inside the moving rocket, events A and B (the light hitting the room's front and back) are simultaneous because both light waves travel at the same speed towards each end.
• For the Earth-based observer, these events are not simultaneous. The relative motion of the rocket affects how this observer perceives the timing of events A and B.

This difference happens because time and space are woven together in a four-dimensional spacetime, changing our typical expectations of simultaneity.

Observer Perspective in Relativity

Relativity places a strong emphasis on understanding how different observers perceive events. The observer's state of motion affects how they interpret both time and space.
In our rocket example:

• The astronaut is in the same inertial frame as the events inside the rocket room, perceiving events like the light hitting both ends as happening at the same time.
• The Earth observer is in a different inertial frame because the rocket is moving relative to them. Thus, the events appear sequential rather than simultaneous.

This idea underscores the relativity principle: the laws of physics (like the speed of light) are the same for all observers, but interpretations of events differ depending on the observer's frame of reference.